Vol. 31 No. 2 (2021)

High-dimensional Private Quantum Channels and Regular Polytopes

Junseo Lee
School of Electrical and Electronic Engineering, Yonsei University, Seoul 03722, Korea
Kabgyun Jeong
Research Institute of Mathematics, Seoul National University, Seoul 08826, Korea and School of Computational Sciences, Korea Institute for Advanced Study, Seoul 02455, Korea

Published 10-05-2021

How to Cite

Lee, J., & Jeong, K. (2021). High-dimensional Private Quantum Channels and Regular Polytopes. Communications in Physics, 31(2), 189. https://doi.org/10.15625/0868-3166/15762


As the quantum analog of the classical one-time pad, the private quantum channel (PQC) plays a fundamental role in the construction of the maximally mixed state (from any input quantum state), which is very useful for studying secure quantum communications and quantum channel capacity problems. However, the undoubted existence of a relation between the geometric shape of regular polytopes and private quantum channels in the higher dimension has not yet been reported. Recently, it was shown that a one-to-one correspondence exists between single-qubit PQCs and three-dimensional regular polytopes (i.e., regular polyhedra). In this paper, we highlight these connections by exploiting two strategies known as a generalized Gell-Mann matrix and modified quantum Fourier transform. More precisely, we explore the explicit relationship between PQCs over a qutrit system (i.e., a three-level quantum state) and regular 4-polytopes. Finally, we attempt to devise a formula for such connections on higher dimensional cases.


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